Grothendieck-polynomial rule via modified Kohnert diagrams

Let $w\in S_n$ and $\gamma\in\mathbb{Z}_{\geq 0}^n$. Let $\overline{{\sf KKoh}}(w,\gamma)$ denote the modified family of diagrams associated with $w$ and weight $\gamma$, and let $D(w)$ be the diagram of $w$. Modified Kohnert-diagram rule. One has

$$g_{w,\gamma}=\#\overline{{\sf KKoh}}(w,\gamma),$$

and consequently

$${\mathfrak{G}_w}= \sum_{D\in \overline{{\sf KKoh}}(D(w))}(-1)^{\#D-\ell(w)}x^{{\tt wt}(D)}.$$

This is the paper's replacement for the refuted Ross–Yong rule, but the supplied material gives no resolution status beyond the statement itself.